Integration of Trigonometric Functions
Integration of Trigonometric Functions
Page 1
Page 2
Example 1:
$$\int \sin^7 x \cos x\, dx $$
Solution:
\begin{align*}
\int \sin^7 x \cos x\, dx &= \int (\sin x )^7\cos x \, xdx \\\\
\textup{Let}\,\, u & =\sin x \\
du & = \cos x \, dx \\\\
\int \sin^7 x \cos x\, dx & = \int u^7 \,du \\
& = \frac{u^{7+1}}{7+1}+C \\
& = \frac{u^{8}}{8}+C \\
& = \frac{1}{8} \, u^8+C \\
& = \frac{1}{8} \, (\sin x)^8+C \\
& = \frac{1}{8} \, \sin^8 x+C \\
\end{align*}
Example 2:
$$\int \cos^4 x \sin x\, dx $$
Solution:
\begin{align*}
\int \cos^4 x \sin x\, dx &= \int (\cos x )^4 \sin x \, xdx \\\\
\textup{Let}\,\, u & =\cos x \\
du & =-\sin x \, dx \\
-du & = \sin x \, dx \\\\
\int \cos^4 x \sin x\, dx & = \int u^4 \, (-du) \\
& = -\int u^4 \,du\\
& = -\frac{u^{4+1}}{4+1}+C \\
& = -\frac{u^{5}}{5}+C \\
& = -\frac{1}{5} \, u^5+C \\
& = -\frac{1}{5} \, (\cos x)^5+C \\
& = -\frac{1}{5} \, \cos^5 x+C \\
\end{align*}